Note on cohomology rings of spherical varieties and volume polynomial

TL;DR

This paper describes the cohomology rings of projective spherical G-varieties using volume polytopes, generalizing known results for toric and flag varieties, and explores properties of moment and string polytopes.

Contribution

It provides a unified polytope-based description of cohomology rings for spherical varieties, extending the Khovanskii-Pukhlikov approach beyond toric varieties.

Findings

Unified description for cohomology rings of flag and toric varieties

Description of cohomology ring of complete conics

Proved additivity of the moment polytope for symmetric varieties

Abstract

Let G be a complex reductive group and X a projective spherical G-variety. Moreover, assume that the subalgebra A of the cohomology ring H^*(X, R) generated by the Chern classes of line bundles has Poincare duality. We give a description of the subalgebra A in terms of the volume of polytopes. This generalizes the Khovanskii-Pukhlikov description of the cohomology ring of a smooth toric variety. In particular, we obtain a unified description for the cohomology rings of complete flag varieties and smooth toric varieties. As another example we get a description of the cohomology ring of the variety of complete conics. We also address the question of additivity of the moment and string polytopes and prove the additivity of the moment polytope for complete symmetric varieties.